Harmonic exponential terms are polynomial
Tyler Borgard, Chris Miller · Canadian Mathematical Bulletin · 2025
Abstract Let n be a positive integer and f belong to the smallest ring of functions upper R Superscript n Baseline right arrow double struck upper R $\mathbb R^n\to \mathbb R$ R n → R that contains all real polynomial functions of n variables and is closed under exponentiation. Then there exists d element of double struck upper N $d\in \mathbb N$ d ∈ N such that for all m element of StartSet 0 comma midline horizontal ellipsis comma n EndSet $m\in \{0,\dots , n\}$ m ∈ { 0 , ⋯ , n } and c element of upper R Superscript m $c\in \mathbb R^{m}$ c ∈ R m , if x right arrow from bar f left parenthesis c comma x right parenthesis colon upper R Superscript n minus m Baseline right arrow double struck upper R $x\mapsto f(c,x)\colon \mathbb R^{n-m}\to \mathbb R$ x ↦ f ( c , x ) : R n − m → R is harmonic, then it is polynomial of degree at most d . In particular, f is polynomial if it is harmonic.